33
Acta Math. Univ. Comenianae
Vol. LXV, 1(1996), pp. 33–51
ON THE STRUCTURE OF THE SOLUTION
SET OF EVOLUTION INCLUSIONS WITH
TIME–DEPENDENT SUBDIFFERENTIALS
N. S. PAPAGEORGIOU and F. PAPALINI
Abstract. In this paper we consider evolution inclusions driven by a time dependent subdifferential operator and a set-valued perturbation term. First we show
that the problem with a convex-valued, h∗ -u.s.c. orientor field (i.e. perturbation
term) has a nonempty solution set which is an Rδ -set in C(T, H), in particular
then compact and acyclic. For the non convex problem (i.e. the orientor field is
non convex-valued), without assuming that the functional ϕ(t, x) of the subdifferential is of compact type, we show that for every initial datum ξ ∈ domϕ(0, ·) the
solution set S(ξ) is nonempty and we also produce a continuous selector for the
multifunctions ξ → S(ξ). Some examples of distributed parameter systems are also
included.
1. Introduction
In a recent paper (cf. [21]) we established the nonemptiness and path-connectedness of the solution set of an evolution inclusion driven by a time-dependent subdifferential ∂ϕ(t, x) and a nonconvex-valued, h-Lipschitz in x multivalued perturbation term F (t, x).
In this paper first we consider time-dependent subdifferential evolution inclusions with a convex valued perturbation term F (t, x) satisfying a more general
continuity hypothesis in the x-variable. For such evolution inclusions we show
that the solution set is Rδ in C(T, H), in particular then nonempty, compact and
connected. Then we return to the nonconvex problem with an h-Lipschitz in x
perturbation term F (t, x). Without assuming ϕ(t, ·) is of compact type (which
in the time invariant case means that ∂ϕ(·) generates a compact semigroup of
contractions), we establish the nonemptiness of the solution set S(ξ) and in addition we generate a continuous selector for the multifunction ξ 7→ S(ξ). Finally we
present some examples of parabolic distributed parameter systems illustrating the
applicability of our results.
Received July 27, 1995.
1980 Mathematics Subject Classification (1991 Revision). Primary 34G20; Secondary 35K55.
Key words and phrases. h∗ -upper semicontinuity, Hausdorff metric, subdifferential, function
of compact type, parabolic problem, compact embedding.
34
N. S. PAPAGEORGIOU and F. PAPALINI
The problem of connectedness of the solution set of differential inclusions in
RN was investigated by many authors. The first to establish that this set is Rδ
in C(T, RN ) were Himmelberg-Van Vleck (cf. [12]) for autonomous systems and
De Blasi-Myjak (cf. [7]) for nonautonomous systems. The result of De Blasi-Myjak
was extended to systems with state constraints by Hu-Papageorgiou (cf. [14]). We
should also mention the remarkable recent work of De Blasi-Pianigiani (cf. [8], [9])
who developed the so-called “Baire category method” to study the structure of the
extremal solutions of differential inclusions in RN and in Banach spaces. However
their compactness and continuity hypotheses preclude the applicability of their results to systems involving unbounded operators (i.e. distributed parameter systems
with multivalued terms). There is also the very recent work of De Blasi-PianigianiStaicu (cf. [10]) which deals with semilinear evolution inclusion monitored by a
time invariant unbounded linear operator generating a C0 -semigroup. Extension
of the results of De Blasi-Pianigiani-Staicu can be found in Hu-LakshmikanthamPapageorgiou (cf. [13]). We should also mention the work of Ballotti (cf. [1]) which
deals with semilinear evolution equation driven by a time invariant linear operator
which is the generator of a C0 -semigroup and by a single-valued perturbation term
f (t, x) which is jointly continuous. Our work here extends that of Ballotti. Finally
on the question of existence of continuous selectors for the solution multifunction
ξ 7→ S(ξ) there are the works of Cellina (cf. [5]) for differential inclusions in RN
and of Staicu (cf. [23]) for evolution inclusions driven by a time invariant maximal
monotone operator on a Hilbert space.
2. Mathematical Preliminaries
Let X be a separable Banach space. We will be using the following notations:
Pf (c) (X) = {A ⊆ X : A nonempty, closed and (convex)},
P(w)k(c) (X) = {A ⊆ X : A nonempty, (weakly-)compact (convex)}.
If (Ω, Σ, µ) is a finite measure space, a multifunction F : Ω → Pf(X) is said to
be measurable,
if for all x ∈ X, the function ω 7→ d(x, F (ω))
= inf kx − zk : z ∈
F (ω) is measurable. If F (·) is measurable, then Gr F = (ω, x) ∈ Ω × X : x ∈
F (ω) ∈ Σ×B(X), with B(X) being the Borel σ-field of X (graph measurability),
while the converse is true if Σ is µ-complete. By SFp (1 ≤ p ≤ ∞) we will denote
the set of all measurable selectors
of F (·) that belong in the Lebesgue-Bochner
space Lp (Ω, X); i.e. SFp = f ∈ Lp (Ω, X) : f (ω) ∈ F (ω) µ-a.e. . In general this
set may be empty. It is easy to check using Aumann’s selection Theorem (cf. [24,
Theorem 5.10]), that for a graph measurable multifunction F : Ω → 2X \ {∅},
SFp is nonempty if and only if the function ω 7→ inf{kzk : z ∈ F (ω)} belongs
to Lp (Ω, R+ ). Recall that a subset K of Lp (Ω, X) is decomposable if for every
triple (f, g, A) ∈ K × K × Σ, we have f χA + gχAc ∈ K, where χA denotes the
characteristic function of the set A. Clearly SFp is decomposable.
STRUCTURE OF SOLUTION SET
35
A subset A of X is said to be a absolute retract if, given any metric space Y and
a closed B ⊆ Y and a continuous function f : B → A, there exists a continuous
extensions f˜: Y → A of f . Then A is said to be a Rδ -set if A = ∩n≥1 An for a
decreasing sequence of compact absolute retracts An of X (cf. [15]). Every Rδ -set
is acyclic.
Recall that on Pf (X) we define a generalized metric, known in the literature as
the “Hausdorff metric”, by setting, for A, B ∈ Pf (X),
h(A, B) = max sup{d(a, B) : a ∈ A}, sup{d(b, A) : b ∈ B}
(where d(a, B) = inf{ka − bk : b ∈ B}; similarly for d(b, A)). A multifunction
F : T → Pf (X) is said to be Hausdorff continuous (h-continuous) if it is continuous
from T into the metric space (Pf (X), h). Moreover, F is said to be Hausdorff upper
semicontinuous (h∗ -u.s.c.) if, for every t ∈ T and for all ε > 0 there exists δ > 0,
such that |t − t0 | < δ ⇒ F (t0 ) ⊆ F (t) + εB1 , where B1 is the unit ball in X.
Let ϕ : X → R = R ∪ {+∞}. We will say that ϕ(·) is proper, if it not identically
+∞. Assume that ϕ(·) is proper, convex and l.s.c. (usually this family of R-valued
functions is denoted by Γ0 (X)). By dom ϕ we will denote the effective domain of
ϕ(·); i.e. dom ϕ = {x ∈ X : ϕ(x) < ∞}. The subdifferential of ϕ(·) at x, is the set
∂ϕ(x) = {x∗ ∈ X ∗ : (x∗ , y − x) ≤ ϕ(y) − ϕ(x), ∀ y ∈ dom ϕ}
(in this definition, by (·, ·) we denote the duality brackets for the pair (X, X ∗ )).
It is well-known that if ϕ(·) is Gateaux differentiable at x, then ∂ϕ(x) = {ϕ0 (x)}.
We say that ϕ ∈ Γ0 (X) is of compact type, if for all λ ∈ R+ , the level set
{x ∈ X : ϕ(x) + kxk2 ≤ λ} is compact.
Let T = [0, b] and H a separable Hilbert space. We consider the following
multivalued Cauchy problem:
(1)
−ẋ(t) ∈ ∂ϕ(t, x(t)) + F (t, x(t)), a.e. on T, x(0) = ξ, ξ ∈ dom ϕ(0, ·).
By a solution of (1) we mean a function x ∈ C(T, H) such that x(·) is absolutely
continuous on any closed subinterval of (0, b) and with the property:
1) x(t) ∈ dom ϕ(t, ·), a.e. on T ;
2) ∃ f ∈ L2 (T, H) such that f (t) ∈ F (t, x(t)) and −ẋ(t) ∈ ∂ϕ(t, x(t)) + f (t), a.e.
on T ;
3) x(0) = ξ.
Recall that an absolutely continuous functions x: (0, b) → H is differentiable
almost everywhere (see [2, Theorem 2.1, p. 16]) and so in problem (1) the derivative
ẋ(·) is a strong derivative.
Following S. Yotsutani (cf. [26]) we make the following hypothesis on the function ϕ(t, x), which will be in effect throughout this paper:
H(ϕ):
ϕ : T × X → R ∪ {+∞} is a function such that
36
N. S. PAPAGEORGIOU and F. PAPALINI
(i) ∀ t ∈ T , x 7→ ϕ(t, x) is proper, convex, l.s.c;
(ii) for every integer r > 0, there exist Kr > 0, an absolutely continuous function
gr : T → R with ġr ∈ Lβ (T ) and a function of bounded variation hr : T → R
such that if t ∈ T , x ∈ dom ϕ(t, ·) with kxk ≤ r and s ∈ [t, b], then there
exists x̂ ∈ dom ϕ(s, ·) satisfying
kx − x̂k ≤ |gr (s) − gr (t)|(ϕ(t, x) + Kr )α
and
ϕ(s, x̂) ≤ ϕ(t, x) + |hr (s) − hr (t)|(ϕ(t, x) + Kr )
where α ∈ [0, 1] and β = 2 if α ∈ 0, 12 or β = 1/(1 − α) if α ∈ 12 , 1 .
Remark. Hypothesis H(ϕ)(ii) allows the effective domain dom ϕ(t, ·) of ϕ(t, ·)
to vary in a regular way with respect to t ∈ T without excluding the possibility
that dom ϕ(t, ·) ∩ dom ϕ(s, ·) = ∅ if t 6= s. This hypothesis, which is suitable for
the analysis of obstacle problems, has its origin (in more restrictive form) in the
works of Kemnochi (cf. [16]) and Yamada (cf. [25]).
3. Topological Structure of the Solution Set
Our first result establishes the nonemptiness and the topological structure of
the solution set S(ξ) ⊆ C(T, H) of (1). We will need the following hypothesis on
the orienter field F :
H(F ):
F : T × H → Pf c (H) is a multifunction such that
(i) ∀ x ∈ H, t 7→ F (t, x) is measurable;
(ii) ∀ t ∈ T , x 7→ F (t, x) is h∗ -u.s.c.;
(iii) ∃ a, c ∈ L2 (T, R+ ):
kF (t, x)k = sup kzk : z ∈ F (t, x) ≤ a(t) + c(t)kxk, a.e. in T, ∀ x ∈ H.
In the proof of the structural theorem concerning S(ξ) we will need some auxiliary results which we state next.
The first is an approximation lemma which can be proved as Proposition 4.1.
of De Blasi (cf. [6]), with some appropriate modifications to accomodate for the
presence of t ∈ T .
Lemma 1. Let F : T × X → Pf c (H) be a multifunction such that
(i) ∀ x ∈ H, t 7→ F (t, x) is measurable;
(ii) ∀ t ∈ T , x 7→ F (t, x) is h∗ -u.s.c.;
(iii) ∃ ψ ∈ L2 (T, R+ ) : kF (t, x)k ≤ ψ(t), a.e. in T , ∀ x ∈ H;
37
STRUCTURE OF SOLUTION SET
then there exists a sequence of multifunctions Fn : T × H → Pf c (H), n ≥ 1, with
the properties:
I) ∀ n ≥ 1 and ∀ x ∈ H there exist kn (x) > 0 and εn > 0 such that if
x1 , x2 ∈ B εn (x) = {y ∈ H : kx − yk ≤ εn }, then h(Fn (t, x1 ), Fn (t, x2 )) ≤
kn (x)ψ(t)kx1 − x2 k a.e. on T (i.e. Fn (t, ·) is locally h-Lipschitz);
II) F (t, x) ⊆ · · · ⊆ Fn (t, x) ⊆ Fn+1 (t, x) ⊆ . . . , ∀ (t, x) ∈ T × H;
III) kFn (t, x)k ≤ ψ(t) a.e. in T , ∀ x ∈ H;
h
IV) Fn (t, x) −
→ F (t, x) as n → ∞ for every (t, x) ∈ T × H
and finally there exists maps un : T × H → H, n ≥ 1, measurable in t, locallyLipschitz in x (as Fn (t, ·)) and un (t, x) ∈ Fn (t, x) for every (t, x) ∈ T × H.
Moreover if F (t, ·) is h-continuous, then for every n ≥ 1, t → Fn (t, x) is measurable.
Proof. For each (fixed) n ≥ 1 Sn = b x, 31n x∈H is an open cover of H. Let
Pn = {γxn }x∈H be a locally-Lipschitz partition of unity subordinate to Sn .
Let x0 ∈ H. Then there exist kn (x0 ) and εn > 0 such that
N (n, x0 ) = x ∈ H : supp γxn ∩ B εn (x0 ) 6= ∅
is finite and for every x1 , x2 ∈ B εn (x0 ) we have
X
n
γxnk (x1 ) − γxnnk (x2 ) ≤ kn (x0 )kx1 − x2 k.
xn
k ∈N (n,x0 )
Define
Fn (t, x0 ) =
X
xn
k ∈N (n,x0 )
γxnn (x0 )Gnk (t)
k
where Gnk (t) = co F t, B xnk , 32n . Evidently since the function x 7→ γxn (x0 )
P
n
n
vanish outside the set N
(n, x0 ) we have Fn (t, x0 ) =
x∈H γx (x0 )Gx (t) with
2
n
Gx (t) = co F t, B x, 3n .
Now define
X
un (t, x0 ) =
γxnnk (x0 )fkn (t)
xn
k ∈N (n,x0 )
with fkn (·) being a measurable selector of Gnk (·). It exists since by hypothesis
t 7→ F (t, x) is measurable. Clearly for every x ∈ B εn (x0 ) we have
Fn (t, x) =
X
xn
k ∈N (n,x0 )
γxnn (x)Gnk (t) and un (t, x) =
k
X
xn
k ∈N (n,x0 )
So for every x1 , x2 ∈ B εn (x0 ) and almost all t ∈ T we have
h(Fn (t, x1 ), Fn (t, x2 )) ≤ kn (x0 )ψ(t)kx1 − x2 k
γxnnk (x)fkn (t).
38
N. S. PAPAGEORGIOU and F. PAPALINI
and
kun (t, x1 ) − un (t, x2 )k ≤ kn (x0 )ψ(t)kx1 − x2 k
which establishes the local h-Lipshitzness of the approximating multifunctions
Fn (t, ·), n ≥ 1, and the local Lipshitzness of the sector un (t, ·), n ≥ 1.
Next we will show that {Fn (t, x)}n≥1 is a increasing sequence. Fix y ∈ H
and let M (n, y) (resp. M (n + 1, y))) be the nonempty finite set of all functions
γxn ∈ Pn (resp. γxn+1 ∈ Pn+1 ) whose support contains y. Let 1 ≤ j ≤ |M (n, y)| and
1 ≤ k ≤ |M (n + 1, y)|. We will show that Gn+1
(t) ⊆ Gnxnj (t), t ∈ T . To this end
xn+1
k
n+1
1
observe that since y ∈ supp γxnnj and y ∈ supp γxn+1
, 3n+1
n+1 , we have y ∈ B xk
k
and y ∈ B xnj , 31n .
2
So if z ∈ B xn+1
, 3n+1
we have
k
kz − xnj k ≤ kz − xn+1
k + kxn+1
− yk + ky − xnj k <
k
k
2
3n+1
+
1
3n+1
+
1
2
= n
n
3
3
2
⊆ B xnj , 32n which in turn implies that Gn+1
and so B xn+1
, 3n+1
(t) ⊆ Gnxnj (t),
k
xn+1
k
t ∈ T . From this fact and the definition of Fn (t, y) we immediately deduce that
Fn (t, y) ⊆ Fn+1 (t, y). In a similar fashion we can also get that F (t, y) ⊆ Fn (t, y)
for every n ≥ 1 and every (t, y) ∈ T × H.
h
Now we will show that Fn (t, x) −
→ F (t, x) as n → ∞ for every (t, x) ∈ T × H.
Since by hypothesis F (t, ·) is h∗ -u.s.c. given ε > 0 we can find δ(t, x, ε) > 0 such
that F (t, y) ⊆ F (t, x) + 3ε B1 for every y ∈ H with kx − yk ≤ δ.
Let N0 (t, x, ε) ≥ 1 be such that for n ≥ N0 we have 31n ≤ δ3 . Let v ∈ H be such
that γvn ∈ M (n, x) and let y ∈ B v, 32n . For n ≥ N0 we have
kx − yk ≤ kx − vk + kv − yk <
2
1
1
+
= n−1 ≤ δ.
3n 3n
3
Thus for n ≥ N0 , for v ∈ H for which γvn ∈ M (n, x) and for y ∈ B v, 32n we
have F (t, y) ⊆ F (t, x) + 3ε B1 . So Gnv (t) ⊆ F (t, x) + 3ε B1 ⊆ F (t, x) + 2ε
3 B1 for
all n ≥ N0 and all v ∈ H such that γvn ∈ M (n, x). Thus we get that Fn (t, x) ⊆
h
F (t, x) + 2ε
→
3 B1 ⊆ F (t, x) + εB1 for n ≥ N0 and so we conclude that Fn (t, x) −
F (t, x) as n → ∞.
Finally if F (t, ·) is h-continuous, then for {ym }m≥1 a dense subset of B xnk , 32n
we have that Gnk (t) = co (∪m≥1 F (t, ym )). Invoking Proposition 2.3 and Theorem 9.1 of [11], we deduce that t 7→ Gnk (t) is measurable and so t 7→ Fn (t, x) is
measurable.
Now let p : L2 (T, H) → C(T, H) be the map which to each g ∈ L2 (T, H) assigns
the unique solution of
−ẋ(t) ∈ ∂ϕ(t, x(t)) + g(t), a.e. on T, x(0) = ξ ∈ dom ϕ(0, ·).
STRUCTURE OF SOLUTION SET
39
The existence (and of course uniqueness) of the solution of the above Cauchy
problem follows from the result of [26].
We have the following result concerning the solution map p(·).
Lemma 2. If hypothesis H(ϕ) holds, if for every t ∈ T ϕ(t, ·) is of compacttype and ξ ∈ dom ϕ(0, ·), then p(·) is completely continuous (hence compact).
Proof. Let gn → g weakly in L2 (T, H). For economy in the notation set xn =
p(gn ), n ≥ 1, and x = p(g). Exploiting the monotonicity of the subdifferential we
get
(−ẋn (t) + ẋ(t), x(t) − xn (t)) ≤ (gn (t) − g(t), x(t) − xn (t)) a.e. on T,
1 d
⇒
kxn (t) − x(t)k2 ≤ kgn (t) − g(t)k kxn (t) − x(t)k a.e. on T,
2 dt
Z t
1
2
⇒
kxn (t) − x(t)k ≤
kgn (s) − g(s)k kxn (s) − x(s)k ds, ∀ t ∈ T.
2
0
Using Lemma A.5, p. 157, of [4], we get that
Z
kxn (t) − x(t)k ≤
0
t
kgn (s) − g(s)k ds,
∀ t ∈ T.
Since gn → g weakly in L2 (T, H) we can find M1 > 0 such that kgn k1 , kgk1 ≤
M1 , n ≥ 1. So for all n ≥ 1 and all t ∈ T we have
kxn (t)k ≤ kxk∞ + 2M1 = M2 < ∞ .
Moreover from inequality (7.9), p. 645 of [26], we see that there exists M3 > 0
such that for all t ∈ T and all n ≥ 1 we have
ϕ(t, xn (t)) ≤ M3 ,
(in fact M3 depends only on the total variation of hM2 (·), on kgM2 kβ , on ξ, on
ϕ(0, ξ) and on M1 ; see [26]). So for every t ∈ T we have
{xn (t)}n≥1 ⊆ y ∈ H : kyk2 + ϕ(t, y) ≤ M22 + M3 = M4
⇒ {xn (t)}n≥1 ∈ Pk (H) (recall that ϕ(t, ·) is of compact-type).
Also if s, t ∈ T , s ≤ t we have
Z t
Z t
√
kxn (t) − xn (s)k = ẋn (τ ) dτ kẋn (τ )k dτ ≤
t − s sup kẋn k2 .
≤
s
s
But from inequality (7.5) p. 645 of [26], we know that sup kẋn k2 = M5 < +∞
with M5 > 0 depending only on M1 , kξk and ϕ(0, ξ). Thus we deduce that {xn }n≥1
40
N. S. PAPAGEORGIOU and F. PAPALINI
is equicontinuous. Invoking the Arzela-Ascoli theorem we get that {xn }n≥1 is
relatively compact in C(T, H). By passing to a subsequence if necessary, we may
assume that xn → y in C(T, H) and ẋn → ẏ weakly in L2 (T, H).
Let Φ : L2 (T, H) → R = R ∪ {+∞} be defined by
Rb
Φ(x) =
0
ϕ(t, x(t)) dt,
+∞,
if ϕ(·, x(·)) ∈ L1 (T )
otherwise
(note that by Lemma 3.4, p. 629 of [26], for every x ∈ L2 (T, H), t →
7 ϕ(t, x(t))
is measurable, moreover Corollary 4.1, p. 633 tell us that dom Φ =
6 ∅). It is
well-known (see for example [26, Lemma 4.4, p. 634]) that
∂Φ(x) = v ∈ L2 (T, H) : v(t) ∈ ∂ϕ(t, x(t)) a.e. on T .
For every n ≥ 1 [xn , −ẋn − gn ] ∈ Gr ∂Φ since ∂Φ(·) is a maximal monotone
operator, Gr ∂Φ is demiclosed. Therefore [y, −ẏ − g] ∈ Gr ∂Φ and so −ẏ(t) ∈
∂ϕ(t, y(t)) + g(t) a.e. on T , y(0) = ξ, i.e. y = p(g) = x. So we conclude that
xn → x in C(T, H), proving that indeed p(·) is completely continuous.
Now observe that if x1 , x2 ∈ C(T, H) are solutions of
−ẋ(t) ∈ ∂ϕ(t, x(t)) + g(t), a.e. on T, x(0) = ξ1 ∈ dom ϕ(0, ·),
−ẋ(t) ∈ ∂ϕ(t, x(t)) + g(t), a.e. on T, x(0) = ξ2 ∈ dom ϕ(0, ·),
respectively, then kx1 (t) − x2 (t)k ≤ kξ1 − ξ2 k, ∀ t ∈ T . This is an immediate
consequence of the monotonicity of the subdifferential operator.
If p̂ : dom ϕ(0, ·) × L2 (T, H) → C(T, H) is the map which to each (ξ, g) ∈
dom ϕ(0, ·) × L2 (T, H) assigns the unique solution of
−ẋ(t) ∈ ∂ϕ(t, x(t)) + g(t), a.e. on T, x(0) = ξ ∈ dom ϕ(0, ·),
(cf. [26]), then we have an alternative version of Lemma 2.
Lemma 2’. If hypothesis H(ϕ) holds, if for every t ∈ T ϕ(t, ·) is of compacttype and if for every ξ ∈ dom ϕ(0, ·) there exist M , r > 0 such that ϕ(0, x) ≤
M , ∀ x ∈ dom ϕ(0, ·) with |x − ξ| < r, then the solution map p̂ is sequentially
continuous by considering on L2 (T, H) the weak topology.
The proof of this lemma is the same as that of the previous, hence is omitted.
Now we are ready for the result on the nonempiness and topological structure
of the solution set S(ξ).
STRUCTURE OF SOLUTION SET
41
Theorem 3. If hypotheses H(ϕ), H(F ) hold, if for every t ∈ T ϕ(t, ·) is of
compact type and ξ ∈ dom ϕ(0, ·), then S(ξ) is a Rδ -set in C(T, H).
Proof. First let us derive an a priori bound for the elements in S(ξ). So let
x ∈ S(ξ) and let y ∈ C(T, H) be the unique solution of
−ẏ(t) ∈ ∂ϕ(t, y(t)), a.e. on T, y(0) = ξ.
Let f ∈ SF2 (·,x(·)) be such that x = p(f ). As before from the monotonicity of the
subdifferential operator we have
(−ẋ(t) + ẏ(t), y(t) − x(t)) ≤ (f (t), y(t) − x(t) a.e. on T,
Z t
1
2
kx(t) − y(t)k ≤
⇒
kf (s)k ky(s) − x(s)k ds, ∀ t ∈ T.
2
0
Once again Lemma A.5, p. 157, of [4], tell us that
Z
Z
t
kf (s)k ds ≤
kx(t) − y(t) ≤
0
Z
⇒ kx(t)k ≤ kyk∞ +
t
(a(s) + c(s)kx(s)k) ds, ∀ t ∈ T
0
t
(a(s) + c(s)kx(s)k) ds, ∀ t ∈ T.
0
Using Gronwall’s lemma, we deduce that there exists M > 0 such that kx(t)k ≤ M
for every t ∈ T and every solutions x ∈ S(ξ). Thus without any los of generality
put ψ(t) = a(t) + c(t)M , ψ ∈ L2 (T, R+ ), we may assume that |F (t, x)| ≤ ψ(t),
a.e. in T , ∀ x ∈ C(T, H) (otherwise replace F (t, x) by F (t, rM (x)) with rM (·)
being the M -radial retraction on H; note that t 7→ F (t, rM (x)) is measurable,
x 7→ F (t, rM (x)) is h∗ -u.s.c. and in addition kF (t, rM (x))| ≤ ψ(t) a.e. on T , with
ψ ∈ L2 (T, R+ )).
Now let Fn : T × H → Pf c (H), n ≥ 1, be a sequence of multifunction as
postulated by Lemma 1. For fixed n ≥ 1 we consider the following multivalued
Cauchy problem
(2)
−ẋ(t) ∈ ∂ϕ(t, x(t)) + Fn (t, x(t)), a.e. on T, x(0) = ξ, ξ ∈ T.
We already known that problem (2) above has a nonempty set Sn (ξ) of solutions
(cf. [21]).
Note that Sn (ξ) ⊆ p(V ), with V = {h ∈ L2 (T, H) : kh(t)k ≤ ψ(t) a.e. in T } and
from Lemma 2 we known that p(V ) is compact in C(T, H). Also if xm ∈ Sn (ξ),
m ≥ 1, and xm → x in C(T, H) as m → ∞ we have that xm = p(fm ) with fm ∈
SF2 n (·,xm (·)) . We may assume that fm → f weakly in L2 (T, H) and f ∈ SF2 n (·,x(·)) .
Moreover Lemma 2 tell us that xm → p(f ) = x in C(T, H). Thus Sn (ξ) is closed
hence compact in C(T, H).
42
N. S. PAPAGEORGIOU and F. PAPALINI
We also claim that, for every n ≥ 1, Sn (ξ) is contractible. Let un (t, x) be the
Caratheordory (in fact locally Lipschitz in x) sector of Fn (t, x) (cf. Lemma 1).
Given r ∈ [0, b) and x ∈ Sn (ξ), let z(r, x)(·) ∈ C(T, H) be the unique solution of
−ż(t) ∈ ∂ϕ(t, z(t)) + un (t, z(t)), a.e. on [r, b], z(r) = x(r).
For r = b z(b, x)(b) = x(b). Define h : T × Sn (ξ) → Sn (ξ) by
h(r, x)(t) =
x(t),
for 0 ≤ t ≤ r,
z(r, x)(t),
for r ≤ t ≤ b.
Evidently h(0, x) = z0 and also h(b, x) = x with z0 ∈ C(T, H) being the unique
solution of
−ż(t) ∈ ∂ϕ(t, z(t)) + un (t, z(t)), a.e. on T, z(0) = ξ.
If we can show that h(·, ·) is continuous we will have established the contractibility
of Sn (ξ) in C(T, H). To this end let {(rm , xm )}m ⊆ T × Sn (ξ), with (rm , xm ) →
(r, x) in T × Sn (ξ). We consider two distinct cases:
Case I: rm ≥ r for every m ≥ 1.
Let vm (t) = h(rm , xm )(t), t ∈ T . Evidently vm ∈ Sn (ξ), m ≥ 1, and so by
passing to a subsequence if necessary, we may assume that vm → v in C(T, H).
From the definition of h(·, ·) we see that for t ∈ [0, r] we have v(t) = x(t). Let
y ∈ C(T, H) be the unique solution of
−ẏ(t) ∈ ∂ϕ(t, y(t)) + un (t, v(t)), a.e. on [r, b], y(r) = v(r).
Let N ≥ 1. Then for all m ≥ N large enough we have that −v̇m (t) ∈ ∂ϕ(t, vm (t))+
un (t, vm (t)) a.e. on [rN , b]. As before via the monotonicity of the subdifferential
operator we have
1 d
ky(t) − vm (t)k2 ≤ kun (t, v(t)) − un (t, vm (t))k ky(t) − vm (t)k a.e. on [rN , b]
2 dt
1
1
⇒
ky(t) − vm (t)k2 ≤ ky(rN ) − vm (rN )k2
2
2
Z t
+
kun (s, v(s)) − un (s, vm (s))k ky(s) − vm (s)k ds .
rN
An application of Lemma A.5 of [4] gives us
Z
ky(t) − vm (t)k ≤ ky(rN ) − vm (rN )k +
t
rN
kun (s, v(s)) − un (s, vm (s))k ds.
STRUCTURE OF SOLUTION SET
43
Passing to the limit as m → ∞, we get that
ky(t) − v(t)k ≤ ky(rN ) − v(rN )k for t ∈ [rN , b].
Note that as N → ∞ we have y(rN ) → x(r) and v(rN ) → v(r) = x(r). Since
N ≥ 1 was arbitrary we conclude that y(t) = v(t) for t ∈ [r, b]. Hence v = h(r, x)
and so h(rm , xm ) → h(r, x) in C(T, H) as m → ∞.
Case II: rm ≤ r for every m ≥ 1.
Keeping the notation introduced in the analysis of case I, we see that v(t) = x(t)
for t ∈ [0, r].
Moreover the same arguments as in case I, given us that
Z
ky(t) − vm (t)k ≤ ky(r) − vm (r)k +
t
r
kun (s, v(s)) − un (s, vm (s))k ds for t ∈ [r, b],
and by passing to the limit as m → ∞
ky(t) − v(t)k ≤ ky(r) − v(r)k for t ∈ [r, b].
But y(r) = x(r) = v(r). So y(t) = v(t) for t ∈ [r, b]. Hence v = h(r, x) and so
again we have h(rm , xm ) → h(r, x) in C(T, H) as m → ∞.
In general we can always find a subsequence {rm }m≥1 satisfying case I or case II.
Thus we have proved the continuity of the map h(·, ·). So, for every n ≥ 1, Sn (ξ) is
compact and contractible in C(T, H). We claim that S(ξ) = ∩n≥1 Sn (ξ). Clearly
S(ξ) ⊆ ∩n≥1 Sn (ξ). Let x ∈ ∩n≥1 Sn (ξ). Then by definition x = p(fn ) with
fn ∈ SF2 n (·,x(·)) , n ≥ 1. Evidently {fn }n≥1 is bounded in L2 (T, H). So by passing
to a subsequence, if necessary, we may assume that fn → f weakly in L2 (T, H).
We known that f ∈ SF2 (·,x(·)), (cf. [18]). So x ∈ S(ξ) and therefore we have
S(ξ) = ∩n≥1 Sn (ξ). Using a result of [15], we conclude that S(ξ) is a Rδ -set
in C(T, H).
An immediate consequence of Theorem 3 above is the following Kneser-type
theorem for (1).
Corollary 4. If hypotheses H(ϕ), H(F ) hold and for every t ∈ T ϕ(t, ·) is of
compact type, then, for every t ∈ T , the set R(t) = S(ξ)(t) = {x(t) : x ∈ S(ξ)}
(the reachable set at time t ∈ T ) is compact and connected in H.
Also a consequence of Lemma 2’ is the following continuity result about multifunction ξ 7→ S(ξ). For this result the following weaker version of hypothesis
H(F ) will suffice.
H(F )1 :
F : T × H → Pf c (H)
is a multifunction such that
44
N. S. PAPAGEORGIOU and F. PAPALINI
(i) ∀ x ∈ H, t 7→ F (t, x) is measurable;
(ii) ∀ t ∈ T , Gr F (t, ·) is sequentially closed in H × Hw ; (here Hw stands for the
Hilbert space H equipped with the weaker topology);
(iii) ∃ a, c ∈ L2 (T, R+ ):
kF (t, x)k = sup{kzk : z ∈ F (t, x)} ≤ a(t) + c(t)kxk, a.e. in T, ∀ x ∈ H.
Proposition 5. If hypotheses H(ϕ), H(F )1 hold and for every t ∈ T ϕ(t, ·) is
of compact type, then S : dom ϕ(0, ·) → Pk (C(T, H)) is u.s.c..
Proof. The set S(ξ) is nonempty for ξ ∈ dom ϕ(0, ·) (see [20] with the obvious
modifications), while the compactness of S(ξ) follows from Lemma 2’ as in the
proof the Theorem 3. Now we need to show that given C ⊆ C(T, H) nonempty
closed, the set S − (C) = {ξ ∈ dom ϕ(0, ·) : S(ξ) ∩ C 6= ∅} is closed in dom ϕ(0, ·) ⊆
H. To this end let ξn ∈ S − (C), n ≥ 1 and assume that ξn → ξ in H, with
ξ ∈ dom ϕ(0, ·). Let xn ∈ S(ξn ) ∩ C, n ≥ 1. For each n ≥ 1 let xn = p(fn ),
fn ∈ SF2 (·,xn (·)) . Since {fn }n is bounded in L2 (T, H) (cf. hypothesis H(F )1 (iii)
by passing to a subsequence if necessary we may assume that fn → f weakly in
L2 (T, H). From Lemma 2’ we have that xn → x in C(T, H) and from hypothesis
H(F )1 (ii) and Theorem 3.1 of Papageorgiou (cf. [18]), we have that f ∈ SF2 (·,x(·)) .
So x ∈ S(ξ) ∩ C, i.e. ξ ∈ S − (C). Therefore S(·) is u.s.c..
Next we will generate a continuous selector for the multifunction ξ 7→ S(ξ). For
this we will need the following hypothesis on the orientor field F (t, x).
H(F )2 :
F : T × H → Pf (H) is a multifunction such that
(i) ∀ x ∈ H, t 7→ F (t, x) is measurable;
(ii) ∃ x ∈ L1 (T, R+ ), such that h(F (t, x), F (t, y)) ≤ k(t)kx − yk a.e. on T , ∀ x, y ∈
H;
(iii) ∃ a, c ∈ L2 (T ):
kF (t, x)k = sup{kzk : z ∈ F (t, x)k ≤ a(t) + c(t)kxk, a.e. in T, ∀ x ∈ H.}
Theorem 6. If hypotheses H(ϕ), H(F )2 hold, then there exists u : dom ϕ(0, ·)
→ C(T, H) a continuous map such that u(ξ) ∈ S(ξ) for every ξ ∈ dom ϕ(0, ·).
Proof. Let x0 (ξ)(·) ∈ C(T, H) be the unique solution of the evolution equation
(cf. [26])
−ẋ(t) ∈ ∂ϕ(t, x(t)), a.e. onT, x(0) = ξ ∈ dom ϕ(0, ·).
Let R0 : dom ϕ(0, ·) → Pf (L1 (T, H)) be defined R0 (ξ) = SF1 (·,x0 (ξ)(·)) . Then
R0 (·) is h-continuous and so we can apply Theorem 3 of [3] and get r0 : dom ϕ(0, ·)
STRUCTURE OF SOLUTION SET
45
→ L1 (T, H) a continuous map such that r0 (ξ) ∈ R0 (ξ) for every ξ ∈ dom ϕ(0, ·).
Let x1 (ξ)(·) ∈ C(T, H) be the unique solution of
−ẋ(t) ∈ ∂ϕ(t, x(t)) + r0 (ξ)(t), a.e., x(0) = ξ.
We claim that by induction we can generate two sequence {xn (ξ)(·)}n≥0 ⊆
C(T, H) and {rn (ξ)(·)}n≥0 ⊆ L2 (T, H), with ξ ∈ dom ϕ(0, ·), satisfying:
(a) xn (ξ)(·) ∈ C(T, H) is the unique solution of
−ẋ(t) ∈ ∂ϕ(t, x(t)) + rn−1 (ξ)(t), a.e., x(0) = ξ, ∀ n ≥ 1;
(b) ξ 7→ rn (ξ) is continuous from dom ϕ(0, ·) into L1 (T, H), ∀ n ≥ 0;
(c) rn (ξ)(t) ∈ F (t, xn (ξ)(t)) a.e. on T , for every ξ ∈ dom ϕ(0, ·), ∀ n ≥ 0;
(d) krn (ξ)(t) − rn−1 (ξ)(t)k ≤ k(t)βn (ξ)(t) a.e. on T , ∀ n ≥ 1,
Rt
(θ(t))n−1
n−1
Pn
ε
with βn (ξ)(t) = 0 λ(ξ)(s) (θ(t)−θ(s))
ds + b
k=0 2k+1
(n−1)!
(n−1)! with ε > 0,
Rt
λ(ξ)(t) = a(t) + c(t)kx0 (ξ)(t)k and θ(t) = o k(s) ds.
Since kx0 (ξ)(t) − x0 (ξ 0 )(t)k ≤ kξ − ξ 0 k for every t ∈ T , ξ, ξ 0 ∈ dom ϕ(0, ·) we see
that ξ 7→ λ(ξ)(·) and ξ 7→ βn (ξ)(·) are continuous from dom ϕ(0, ·) into L1 (T, H).
Suppose we were able to produce {xk (ξ)}nk=0 and {rk (ξ)}nk=0 satisfying (a)–(d)
above.
Let xn+1 (ξ)(·) ∈ C(T, H) be the unique solution of
−ẋ(t) ∈ ∂ϕ(t, x(t)) + rn (ξ)(t), a.e. on T, x(0) = ξ ∈ dom ϕ(0, ·).
As before because of the monotonicity of the subdifferential operator and using
Lemma 1.5 of [4], we get
Z t
(∗) kxn+1 (ξ)(t) − xn (ξ)(t)k ≤
krn (ξ)(s) − rn−1 (ξ)(s)k ds
Z
0
t
≤
k(s)βn (ξ)(s)
0
Z
Z
t
s
(θ(s) − θ(τ ))n−1
λ(ξ)(τ )
dτ ds
(n − 1)!
0
0
!
Z t
n
X
ε
(θ(s))n−1
k(s)
ds
+b
2k+1
(n − 1)!
0
k=0
!
Z t
Z t
n
X
(θ(τ ) − θ(s))n−1
ε
(θ(t))n
=
λ(ξ)(s)
k(τ )
dτ ds + b
k+1
(n − 1)!
2
n!
0
s
k=0
!
Z t
Z t
n
X
d (θ(τ ) − θ(s))n
ε
(θ(t))n
=
λ(ξ)(s)
dτ ds + b
n!
2k+1
n!
0
s dτ
k=0
!
Z t
n
X
(θ(t) − θ(s))n
ε
(θ(t))n
=
λ(ξ)(s)
ds + b
n!
2k+1
n!
o
=
k(s)
k=0
< βn+1 (ξ)(t) a.e.
46
N. S. PAPAGEORGIOU and F. PAPALINI
Using hypothesis H(F )2 (ii) we have
(∗∗)
d(rn (ξ)(t), F (t, xn+1 (ξ)(t)) ≤ k(t)kxn (ξ)(t) − xn+1 (ξ)(t)k
≤ k(t)βn+1 (ξ)(t) a.e. on T
Let Rn+1 : dom ϕ(0, ·) → P (L1 (T, H)) be the multifunction defined by
n
o
Rn+1 (ξ) = z ∈ Sf1(·,xn+1(ξ)(·)) : kz(t) − rn (ξ)(t)k ≤ k(t)βn+1 (ξ)(t) a.e. .
From (∗∗) above we know that the multifunction
Γn+1 : dom ϕ(0, ·) → Pwkc (L1 (T, H)),
defined by Γn+1 (ξ)(t) = {v ∈ F (t, xn+1 (ξ)(t)) : kv − rn (ξ)(t)k < k(t)βn+1 (ξ)(t)} ,
is such that Γn+1 (ξ)(t) 6= ∅ a.e. on T .
By modifying the above multifunction on a Lebesgue-null of T , we may assume
without any loss of generality that Γn+1 (ξ)(t) 6= ∅ for every t ∈ T . Also from
Theorem 3.3 of [19] we know that t 7→ F (t, xn+1 (ξ)(t)) is measurable (hence
graph measurable), while (t, v) 7→ kv − rn (ξ)(t)k − k(t)βn+1 (ξ)(t) = γn+1 (ξ)(t, v)
is clearly jointly measurable. So
Gr Γn+1 (ξ) = (v, t) ∈ Gr F (·, xn+1 (ξ)(·)) : γn+1 (ξ)(t) < 0 ∈ L(T ) × B(H)
with L(T ) being the Lebesgue σ-field of T . Apply Aumann’ selection theorem
(cf. [24]), to get z : T → H measurable such that z(t) ∈ Γn+1 (ξ)(t), t ∈ T , so
z(·) ∈ Rn+1 (ξ). Therefore ξ 7→ Rn+1 (ξ) is l.s.c. with decomposable values. Apply
Theorem 3 of [3] to get rn+1 : dom ϕ(0, ·) → L1 (T, H) a continuous map such that
rn+1 (ξ) ∈ Rn+1 (ξ) for every ξ ∈ dom ϕ(0, ·). Hence
rn+1 (ξ)(t) ∈ F (t, xn+1 (ξ)(t)) a.e. and krn+1 (ξ)(t) − rn (ξ)(t)k ≤ k(t)βn+1 (ξ)(t),
a.e. on T . Thus by induction we have produced the two sequences {xn (ξ)}n≥1 ⊆
C(T, H) and {rn (ξ)}n≥1 ⊆ L2 (T, H) satisfying (a)–(d) above.
Then using (∗) we have
Z b
Z b
(θ(b))n
(θ(b) − θ(s))n
krn (ξ)(t) − rn−1 (ξ)(t)k dt ≤
λ(ξ)(s)
ds + bε
n!
n!
0
0
n
(θ(b))
≤
[kλ(ξ)k1 + bε] .
n!
Since ξ 7→ λ(ξ) is continuous from H into L2 (T, H) is locally bounded. So from
the above inequality we deduce that {rn (ξ)}n≥1 is a L1 (T, H)-Cauchy sequence,
locally uniformly in ξ ∈ dom ϕ(0, ·). Also from (a) we have
kxn+1 (ξ) − xn (ξ)kC(T,H) ≤ krn (ξ) − rn−1 (ξ)kL1 (T,H)
⇒ {xn (ξ)}n≥1 is Cauchy in C(T, H), locally uniformly in ξ ∈ dom ϕ(0, ·).
STRUCTURE OF SOLUTION SET
47
let n → ∞. We have xn+1 (ξ) → x(ξ) in C(T, H), rn (ξ) → r(ξ) in L1 (T, H) and
both limits are continuous in ξ ∈ dom ϕ(0, ·). Let y(ξ) ∈ C(T, H) be the unique
solution of
−ż(t) ∈ ∂ϕ(t, z(t)) + r(ξ)(t) a.e. z(0) = ξ ∈ dom ϕ(0, ·).
Because of hypothesis H(F )2 (ii) we have r(ξ)(t) ∈ F (t, x(ξ)(t)) a.e.. As before
we have
Z
kxn (ξ)(t) − y(ξ)(t)k ≤
t
0
krn−1 (ξ)(s) − r(ξ)(s)k ds, t ∈ T
⇒ xn (ξ) → y(ξ) in C(T, H)
⇒ x(ξ) = y(ξ), ξ ∈ dom ϕ(0, ·).
Therefore u : ξ 7→ x(ξ) is the desired selector of ξ 7→ S(ξ).
Remark. Note that Theorem 4 gives as an existence result for (1) without
assuming that ϕ(t, ·) is of compact type. However on the other hand F (t, ·) is
h-Lipschitz. So there is a trade off of hypotheses between ϕ(t, x) and F (t, x).
An immediate consequence of this theorem is the following corollary
Corollary 7. If hypotheses H(ϕ), H(F )2 hold, if there is K ∈ Pf c (H) bounded
such that S(K)(b) ⊆ K, if dom ϕ(b, ·) ⊆ dom ϕ(0, ·) and if dom ϕ(0, ·) ∩ K is
compact in H, then there exists a solution x(·) ∈ C(T, H) for the problem
−ẋ(t) ∈ ∂ϕ(t, x(t)) + F (t, x(t)), a.e. on T, x(0) = x(b).
Proof. Let u : dom ϕ(0, ·) → C(T, H) be the continuous selector of the multifunction ξ 7→ S(ξ) guaranteed by Theorem 4. Let eb : C(T, H) → H be the evaluation map, i.e. eb (x) = x(b). Let û = eb0 u : dom ϕ(0, ·) ∩ K → dom ϕ(0, ·) ∩ K.
This is a continuous and compact map. So Schauder’s fixed point theorem gives
us ξ ∈ dom ϕ(0, ·) ∩ K such that ξ = û(ξ). Then u(ξ)(·) ∈ C(T, H) is the desired
periodic trajectory.
4. Examples
Now let as work out some examples of parabolic distributed parameter systems
to illustrate general abstract results.
Let T = [0, b] and Z ⊆ Rn a bounded domain with smooth boundary Γ. Let
g ∈ W 1,1 (T, W01,p (Z, R)) be given and define
o
n
K(t) = x ∈ W01,p (Z) : x(z) ≥ g(t, z) a.e. on Z .
48
N. S. PAPAGEORGIOU and F. PAPALINI
We consider the following controlled obstacle problem:
 ∂x P
p−2
Dk x(t, z)) ≥ u(t, z) a.e. on Q = T × Z

k=1 Dk (|Dk x(t, z)|
∂t −




x(t, z) ≥ g(t, z) a.e. on Q




p−2
 ∂x − P
Dk x(t, z)) − u(t, z) (x(t, z) − g(t, z)) = 0
k=1 Dk (|Dk x(t, z)|
∂t
(3)
 a.e. on Q






 x|T ×Γ = 0, x(0, z) = x0 (z) a.e. on Z, p ≥ 2,


ku(t, ·)k2 ≤ r(t, x(t, ·)), a.e. on T.
We need the following hypothesis on the function r:
H(r):
r : T × L2 (Z, R) → R+ is a function such that
(i) ∀ x ∈ L2 (Z, R), t 7→ r(t, x) is measurable;
(ii) ∀ t ∈ T , x 7→ r(t, x) is u.s.c.;
(iii) ∃ a, c ∈ L2 (T, R) : r(t, x) ≤ a(t) + c(t)kxk2 , a.e. on t, ∀ x ∈ L2 (Z, R).
Let H = L2 (Z, R) and let ϕ : T × H → R ∪ {+∞} be defined by
 Z
 1
kD(x)(z)kp dz, if x ∈ K(t),
p Z
ϕ(t, x) =

+∞,
otherwise.
We claim that ϕ(t, x) defined above satisfies hypothesis H(ϕ). Indeed let t ≤ s
and let y ∈ K(t). Define x = y − g(t, ·) + g(s, ·). Evidently x ∈ K(s). Also we
have
Z
s
kx − yk2 = kg(s, ·) − g(t, ·)k2 ≤ c1
t
kġ(τ, ·)kp dτ.
In addition through some elementary calculation we have that
|ϕ(t, x) − ϕ(t, y)| ≤ ckDg(s, ·) − Dg(t, ·)kp (kDg(t, ·)kp−1
p
+ kDg(s, ·)kp−1
+ kDykp−1
)
p
p
Z s
≤ c1
kġ(τ, ·)kp dτ(ϕ(t, y) + 1)
t
and so hypothesis H(ϕ) has been satisfied.
Let F : T × H → Pf c (H) be defined by F (t, x) = {u ∈ H : kuk2 ≤ r(t, x)}.
Then it is easy to see that F (t, x) defined above satisfies hypothesis H(F ). Directly
from the definition of subdifferential we have (3) is equivalent to (1) with ϕ(t, x)
and F (t, x) as above. Note that W01,p (Z, R) ,→ L2 (Z, R) compactly and so ϕ(t, ·)is
of compact type. Therefore we can apply Theorem 3 and get:
STRUCTURE OF SOLUTION SET
49
Theorem 8. If hypothesis H(R) holds and x0 ∈ W01,p (Z), x0 ≥ g(0, z) a.e.
on Z, then the solution set of (3) is a Rδ -set in C(T, L2 (Z, R)).
Next consider the following problem with a discontinuous nonlinearity u : Z ×
R → R.
 ∂x PN

 ∂t − k=1 Dj (aij (t, z)Di x(t, z)) ∈ [y(z, x(t, z)), u(z, x(t, z)]
∂x
(4)
∈ β(x(t, z)) a.e. on T × Z,
− ∂n


x(0, z) = x0 (z) a.e. on Z.
We need the following hypotheses:
H(a): for every 1 ≤ i, j ≤ N , aij ∈ L∞ (T × Z, R), aij = aji , there exists c > 0
P
N
such that ckξk2N ≤ N
i,j=1 aij ξi ξj for every ξ = (ξk )k=1 and there exists a
function h : T → R of bounded variation such that |aij (t, z)−aij (s, z)| ≤
|h(t) − h(s)|, ∀ t, s ∈ T , ∀ z ∈ Z.
H(β):
β : R → 2R is a maximal monotone multivalued operator.
Then from Brezis (cf. [4]) we know that there exists j ∈ Γ0 (R) such that β = ∂j.
Assume that j ≥ 0.
Following Rauch (cf. [22]), we set u(z, x) = lim inf y→x u(z, y) and u(z, x) =
lim supy→x u(z, y).
We will make the following hypothesis concerning u and u.
H(U ): 1) the function u, u and u are superpositionally measurable, i.e. for
every x: Z → R measurable, z 7→ u(z, x(z)), u(z, x(z)), u(z, x(z)) are all
measurable,
2) there exist a ∈ L2 (Z, R) and c > 0: |u(z, x)| ≤ a(z) + c|x| a.e. on Z,
∀ x ∈ R.
Let θ ∈ L2 (T, R), θ ≥ 0 and define F0 (t, z, x) = {v ∈ R : θ(t)u ≤ (z, x) ≤
vθ(t)u(z, x)}. Evidently for every x: Z → R measurable, we have that (t, z) →
F0 (t, z, x(z)) is measurable. Moreover since for every (t, z) ∈ T ×Z, x 7→ F0 (t, z, x)
is u.s.c.. Hence if we define F : T × L2 (Z, R) → Pf c (L2 (Z, R)) by F (t, x) =
SF2 0 (t,·,x(·)), we have that F (·, ·) satisfies hypothesis H(F ).
Also let ϕ : T × L2 (Z, R) → R ∪ {+∞} be defined by
Z
 1Z X

a
(t,
z)D
x(z)D
x(z)
dz
+
j(x(v)) dσ(v),

ij
i
j

2 Z

Γ


if x ∈ H 1 (Z, R), j(x(·)) ∈ L1 (Γ, R)
ϕ(t, x) =



+∞,



otherwise.
50
N. S. PAPAGEORGIOU and F. PAPALINI
Let 0 ≤ t ≤ s ≤ b. For every x ∈ dom ϕ(t, ·) = dom ϕ(s, ·) we have
X Z
|ϕ(t, x) − ϕ(s, x)| = (aij (t, z) − aij (s, z))Di x(z)Dj x(z) dz Z
≤ c|h(f ) − h(s)| kDxk22 ≤ c1 |h(t) − h(s)|ϕ(t, x)
(check hypotheses H(a) and H(ϕ)). So we see that H(ϕ) holds. In addition note
that {x ∈ L2 (Z, R) : kxk22 + ϕ(t, x) ≤ λ} is bounded in H 1 (Z, R) and H 1 (Z, R)
embeds compactly in L2 (Z, R). So we deduce that for every t ∈ T , ϕ(t, ·) is of
compact type.
Rewriting (4) in the equivalent abstract from (1) and using Theorem 3, we get:
Theorem 9. If hypotheses H(a), H(β), H(u) hold and x0 (·) ∈ H 1 (Z, R) with
j(x0 (·)) ∈ L1 (Γ, R), then the set of solution (4) is compact and acyclic (thus
connected) in C(T, L2 (Z, R)).
Similarly we can treat the problem:
 ∂x P
− Dj (aij (t, z)Di x(t, z)) + β(x, (t, z))

 ∂t


∈ [u(z, x(t, z)), u(z, x(t, z)], a.e. on T × Z
(5)

x|T ×Γ = 0



x(0, z) = x0 (z) a.e. on Z.
In this case ϕ : T × L2 (Z, R) → R ∪ {+∞} is defined by
Z
 1Z X

a
(t,
z)D
x(z)D
x(z)
dz
+
j(x(z)) dz,

ij
i
j

2 Z

Z


if x ∈ H01 (Z, R), j(x(·)) ∈ L1 (Z, R)
ϕ(t, x) =

 +∞,




otherwise.
As before we can establish the following theorem
Theorem 10. If hypotheses H(a), H(β), H(u) hold and x0 (·) ∈ H01 (Z, R) with
j(x0 (·)) ∈ L1 (Z, R), then the set of solution of (5) is compact and acyclic (thus
connected) in C(T, L2 (Z, R)).
√ ∂x
√
Remark. The problem ∂x
x, ∂n = 0, and ∂x
x, x|T ×Z = 0
∂t −∆x =
∂t −∆x =
are particular cases of (4) and (5) respectively. Therefore our work extends that
of Kikuchi (cf. [17]) and the example in Balloti (cf. [1]).
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N. S. Papageorgiou, National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece
F. Papalini, Department of Mathematics of Perugia University, via Vanvitelli 1, Perugia, 06123
Italy